THE
QUEST OF THE BELL CURVE:

A
CONSTRUCTIONIST APPROACH TO

LEARNING
STATISTICS THROUGH DESIGNING

COMPUTER-BASED
PROBABILITY EXPERIMENTS

Dor Abrahamson Uri
Wilensky

Center for Connected Learning and Computer-Based
Modeling

Northwestern
University

This paper
introduces the rationale, explains the functioning, and describes the process
of developing 'Equidistant Probability', a NetLogo microworld that models
stochastic behavior. In particular, we detail the phases in attempting to
choose suitable parameters and create such graph displays as will permit an
observer to witness the incremental growth of a bell-shaped curve. We argue
that the process of building the model, and in particular the accountability,
motivation, and frustration experienced, were conducive to 'connected learning'
(Wilensky, 1993), through which the design of this microworld is grounded. The
microworld is part of "ProbLab," a suite of Probability-and-Statistics models,
which in turn is part of "Understanding Complexity," a middle-school
curriculum, currently in development.

Introduction

Equidistant Probability[1]
(EP) is a microworld written in NetLogo (Wilensky, 1999)--a multi-agent
parallel-processing modeling environment--as part of an effort of the CCL
(Center for Connected Learning and Computer-Based Modeling,
http://ccl.northwestern.edu) to create software packages that support
middle-school students' learning of probability and statistics. The microworld
and associated activities are intended to draw on students' domain-relevant
personal experience and intuitions (see also Wilensky's, e.g., 1997, Connected
Probability). The design of the EP microworld was done primarily by the first
author (DA) guided by the second author's work on "Connected Probability" and
inspired by Papert's call for increased attention to stochastics (Papert,
1996). The design was informed by a search for an environment that could
pithily convey stochastic behavior as well as the workings of tools for
calculation and display that describe statistical aspects of this behavior. We
wished to create in this environment a suite of models, employing a single
interface, that would address issues of probability from multiple directions.
The result was 'Equidistant Probability', a complex suite of three sub-models:
'Equidistant' (probabilistic), 'Epicenter' ("semi-probabilistic") and
'Circumference' (geometrical-deterministic). In the spirit of constructionism
(see Papert, 1980, 1991; Harel & Papert, 1991), this paper focuses mainly
on DA's own learning through building 'Equidistant' and specifically on his
attempts to run the model so that it would display a specific graph, the
Gaussian bell-shaped or 'normal' distribution curve.

Rationale of Equidistant Probability

The NetLogo interface is a
"patchy" world--a matrix of agent-like square locations--and so lends itself
readily to conceptual metaphors drawing from space and spatiality. Inspired by
discussions of this theme (Noss & Hoyles, 1996; Papert 1996; Wilensky
1996), the EP design started with the attempt to make a connection between
geometry--the science of idealized space--and probability. The thinking began as
follows: From a given patch in the Cartesian patch-matrix (the center of the
circle in Figure 1) a 'turtle'--the Logo agent-creature--"sprouts[2]" at a
random orientation and advances in discrete steps each equal in size to one
patch unit. What is the chance that this walking turtle will land on each of
its adjacent patches, e.g., on the patch directly to its right, as compared to
the patch diagonally above it?

Figure 1: Geometric
determination of the probabilities that a creature sprouting from the middle
of a patch, heading in a random orientation, and stepping 1 unit forward will
land on each of the 8 neighboring patches. The circle is the collection of
all points 1 unit away from the source point and its radii trace steps of
length 1 unit. Note the equilateral triangles (600, 600,
600). The probabilities of turtles landing in a patch are computed
as the ratio between each inferred central angle subtending an arc contained
within a neighboring patch and 3600, i.e. 60/360=1/6 and
30/360=1/12.

( 4 * 1/6 ) + ( 4
* 1/12 ) = 1.00

The thematic problem posed by
the EP microworld becomes more geometrically familiar--relating to vertices of polygons--when one
adds a second turtle that sprouts from a different patch and advances according
to the same set of rules. Given that these two turtles sprout simultaneously
from their respective patches, and march contemporaneously in their discrete
steps, what are the chances that the turtles will land at the same time on the
same patch? Note that when they do land on such a patch, then seeing as they
sprouted and marched simultaneously in equally sized steps, it follows that the
said patch they have now landed on (green patch, in Figure 2, below) is equally
distant from their respective source (red) patches (the green patch's midpoint
is the center of a new circle on the perimeter of which lie the centers of the
source red patches; the distances each turtle has marched from its respective
red patch into the green patch are radii in this circle).

Figure
2: Fragment from the interface of Equidistant Probability showing the
thematic visual metaphor: From each of the (3) selected red patches (the
"vertices") an arrow-shaped pink "creature" sprouts at a random orientation
and darts forward. Creatures sprout simultaneously and advance in discrete
steps. If all (3) creatures meet at a given patch at the same moment, then
that (green) patch is equidistant from the (3) source patches. The patch
label (e.g., 0.22%) indicates

G

R

R

R

the empirical cumulative
frequency of the creatures' repeated rendezvous at that patch. This frequency
will converge stochastically to the geometrically determined expected value
of 0.23% (1/6 * 1/6 * 1/12).

Whereas two source patches
can have many equally distanced patches as rendezvous points for their turtles,
after many attempts it becomes apparent that the turtles tend to co-visit some
of these patches more than they do other patches. Also, more importantly for
this paper, three turtles, too, could find a rendezvous patch, and their
co-visiting of other patches would be rarer. For some configurations of four
(and more) source patches that do have an equidistant point there will be only a single such equidistant
rendezvous patch.

Seeing as the turtles sprout
as "blind mice," oblivious of each other as they are to the science of
statistics, on the sweeping majority of attempts, when all turtles sprout and
head off at random directions, they will not co-visit the same patch. But then again, sometimes
they will. The questions around which EP revolves and which its prospective
users ultimately address is, 'How often, if ever, will the turtles meet on the
equidistant point?' and 'How, if at all, is the turtles' chance of meeting each
other contingent upon their number and upon their relative positions?'

What the nave user does not
initially know but soon discovers is that the "successful attempts"--when all
the active turtles rendezvous on a single patch--are not equally dispersed
across all attempts, just as a '5' does not recur at a rigid schedule across numerous rolls of a die. More
analogously, perhaps, one should speak of the distributed co-occurrence of
three or four 5's when rolling as many dice.

The EP environment allows one
to address fundamental questions of probability, such as 'What does it actually
mean, in practice (situated in Time), that an occurrence has a chance of, say,
1/1296 (1/64)?' Does the probability of an event reflect an
individual observer's confidence level that it will occur as the next outcome,
or does it tell us something about frequencies and limiting values (Hacking,
2001)? How are frequencies to be thought of and used--as mathematical or
empirical (Biehler, 1995)? Why is it that co-occurrences are rarer than individual occurrences?
How does geometrical determinism play against the slowly converging data coming
from the scampering turtles? We suggest that these dichotomous epistemological
and phenomenological aspects of probability--belief vs. frequency, mathematical
vs. empirical, single vs. compound, and deductive vs. inductive--are addressed,
pitted, and connected through the EP design.

The EP microworld is a
laboratory that can scaffold and inform students' habits of research. Moving
back and forth between sub-models, one can use paper-and-pencil and virtual
geometry to pre-determine aspects of posed problems before one runs the model
to test hypotheses, then account for disconfirming evidence that illuminates
back onto one's understanding of geometry (e.g., if one had surmised an 1/8
probability of a creature landing in each of its 8 neighboring patches, one
would then need to account for the variation). Perhaps, most importantly, as
Papert (1996) and many others have noted, the model's computer environment
affords multiple opportunities for running through and processing the outcomes
of a variety of thousands of attempts within a single lesson period (compare
that to the vicissitudes of rolling dice). Using the display affordances of
NetLogo, one can watch how histograms that reflect the distribution of
successful attempts across all attempts grow and take form, such as the
proverbial bell-shaped curve. But would they indeed take that form? DA's
personal learning through building EP was guided by a quest to watch a bell
curve--reflecting 'normal' distribution--coalesce before his eager eyes.

DA's learning can be described as an effort to reconcile
conflicts between his 'psycho-statistics' (Abrahamson & Wilensky,
2003a)--stochastic behavior patterns as interpreted by human perception,
intuition, and experience--and formal mathematics (see also Gigerenzer, 1998;
Biehler, 1995). In particular, these conflicts accounted for difficulty in
modeling the mathematical phenomenon of the bell curve within the EP
environment. The rationale and relevance of the following description is that
mathematics education, and in particular modeling environments, must take such
human biases into account, if we are to create tools for students to negotiate
and connect mathematics to their personal experience (see Wilensky, 1997; also
Abrahamson & Wilensky, 2003b, on S.A.M.P.L.E.R., Statistics As
Multi-Participant Learning-Environment Resource):

The Quest

In his attempt to connect the bell-curve representation
to the EP stochastics experiment, DA's initial instinct was to create code that
would typify and discriminate two distinct classes of events that he was
observing in the experiment outcomes: (1) 'Failure,' when the creatures did not
meet; and (2) 'success' (when the creatures did meet). Next, he was faced with
the task of parsing the succession of experiment outcomes so as to create data
sets that would then be represented as a distribution--a distribution which, he
assumed, would take the form of a bell-curve. But the question was which type
of parsing would lead to the bell-curve distribution?

Let us assume that the string of outcomes in a particular
experiment was as follows, with 'f' standing for 'failure' and 's' standing for
'success':

ffsffffsfsfssffsffffffsfsffsssfsffsfsffs

What should one make of such
a string? Bamberger (1991) speaks of students' spontaneous graphic
representations of sound sequences as modeling and thus revealing the students'
idiosyncratic parsing of the string of auditory stimuli. Likewise, different
parsings of probabilistic events reveal different interpretative underpinnings
of the meaning of probability. It is important to stress that neither of the
following interpretations is "correct" or "incorrect." They are each valid in
their own way (see also "Prob Graphs Basic," part of the CCL "ProbLab" suite of
models).

1."f f s f f f f s f s f s s f f s f
f f f f f s f s f f s s s f s f f s f s f f s." Taken as a string of 40
independent events, one can sum up the number of successes (15) and compute the
probability of a success as the ratio between successes and total outcomes,
i.e. . If the string were
long enough, we could argue for the successes-per-events ratio as being the
limiting value of this phenomenon. Note that such a perspective entirely
ignores the distribution of s's over the string and any variability that could
possibly be observed in this distribution.

2."ffsff ffsfs fssff sffff ffsfs
ffsss fsffs fsffs." Taking a statistical perspective, one may parse the string
into sub-strings of length 5 events each. Now we can compute the probability of
a success occurring in an individual sample: successes
per Sample of length 5 events, or .375 probability of success per single event.
Alternatively, first computing probabilities, one ends up with the same value: .

3."ffs ffffs fs fs s ffs ffffffs fs
ffs s s fs ffs fs ffs." In a random string with a total of n attempts there is an unknown number of sub-strings,
each with a length of 1 through n
and ending with a success. This interpretative parsing of the events
corresponds, perhaps, to an activity in which a success is associated with relief
and momentary discontinuity of the search[3]). Here, the lengths of
the strings are 3, 5, 2, 2, 1, 3, 7, 2, 3, 1, 1, 2, 3, 2, 3. Thus, the average
length of an attempts-until-success sub-string is . Note that whereas in Interpretation #1
we computed (probability of
success per single outcome), here we computed
(average number of attempts until single success). This reciprocity is no
coincidence: In all interpretations, the '40' corresponded to the total number
of events and the '15' corresponds to the total number of successes. However,
each interpretation harbors a different model of the simulated probabilistic
phenomenon, leading to different forms of representation and subsequent statistical
inferences.

Interpretation #3
(samples per success) was DA's intuitive choice for parsing outcomes of the EP
modeled experiment, because as he watched the simulation run in real time, he
interpreted it as the creatures' successive failed attempts to meet that were
each capped by a single success. He graphed the distributed lengths of these
attempt-until-success substrings, and was surprised that this distribution did
not result in a bell-curved distribution, but rather in a Štype curve (Figure 3, on left).

Figure 3: Fragment from Equidistant
Probability. Central distribution is not bell curved because sample (1000
attempts) is too small relative to the frequency (1/432).

Guessing that perhaps instead
of attempts-per-success frequencies
he should rather have graphed the probabilities (the reciprocals of the per-success frequencies), he
created a different graph (Figure 3, on right) but this, too, resulted in a type graph and not in a bell-curve. Only
once he revisited his understanding of 'sample' did DA realize he should parse
strings of outcomes into sub-strings of equal length, tally the successes in
each, and form a distribution of these tallies (Interpretation #2,
above).

But even sampling, DA
found, must be sensitive to the rarity of favored events: His distributions
were coming out lopsided rather than bell-curved (Figure 3, center) because the
samples he was taking were too small, e.g., 1000, for events that occur every
432 attempts, on average. Only once he tweaked the ratio between sample size
and expected frequencies so as to both avoid "floor effects" (ratio is too low,
so many samples have zero successes) and intractability (ratio is too high, so
distribution evolves too slowly), did DA succeed in representing the experiment
outcomes in the form of a bell curve (Figure 4, next page).

Note that DA's insight
into the minimally sufficient sample-to-frequency ratio was informed by his
understanding of the geometrical-determinism in this model. For example, he
expected the frequency of events to converge on 1/432 because he had calculated
the product of 6, 6, and 12, which were, respectively, the chance of each
individual creature to step into the equidistant patch. These complementary
perspectives on stochastics--geometric and probabilistic--converged in DA's process
of making sense of the entire EP suite of models through their design.

Conclusion

Clearly, working in our microworld will be essentially different from
working on it. A designer's productive
learning-through-designing experience is hardly a criterion of a microworld's
efficacy as a learning instrument. Nevertheless, we believe that essential
aspects of DA's learning experience point to a general need in the
Probability-and-Statistics curricula. Therefore, we have designed working in EP
as an exploration that reconstructs the quest of the bell curve or any other
quest. The improved model strives to be sufficiently complex so as to engage
students in both theoretical and empirical probability yet not so complex that
they cannot mathematize and relate these complementary interpretations (see
Henry, 2001). Also, studentswill collaborate in our design and will be able to
share and debate their emergent understandings (Abrahamson, Berland, Unterman,
Shapiro, & Wilensky, 2003), once EP is implemented in several urban middle
and high schools as part of CCL's Statistics-and-Probability model-based
curriculum. Students' collaboration is expected to enhance the epistemological
dialectic designed into EP, e.g., mathematical vs. empirical stochastics,
because conversing students may take different sides as each works on, brings
evidence from, and debates from a different sub-model perspective. Hopefully,
such debate will help students reconcile, as a group and as individuals,
complementary phenomenologies of probability.

The specific arena where such
'complementarity of levels of observation' (Wilensky & Stroup, 2000) acts
out its dialectic is, perhaps, the ambiguous and rival interpretations we bring
to bear while observing the model as it runs. Our anticipation of the model's convergence upon a
particular statistical value, coming from the geometrical proof, espoused with
our real-time perception of the emergent, fuzzy reality, captures the statistical
moment as both chaotic and specific, thus affording the user an environment to
develop a mature, integrative, and 'connected' conceptualization of stochastic
behavior.

Biehler, R. (1995). Probabilistic thinking, statistical reasoning,
and the search of causes: Do we need a probabilistic revolution after we have
taught data analysis? Newsletter of the international study group for
research on learning probability and statistics, 8(1). Accessed December 12, 2002.
http://seamonkey.ed.asu.edu/~behrens/teach/intstdgrp.probstat.jan95.html

[2] "Sprout" is a primitive command in
the NetLogo language. We ask a specific patch to sprout a turtle, and a turtle pops out of the center of
that patch. Likewise, we can ask a plurality of patches to sprout
simultaneously. We use 'sprout' both in the transitive (patch sprouting turtle)
and reflexive (turtle sprouting itself) sense of the verb.